The GitHub Actions build itself takes about ~15 seconds to run. It is generally expected for your site to update around that duration, and take no more than a minute.
Which environment is used to build and deploy the site?
The Actions workflow file publish.yml is configured to pull the latest docker image; you can adjust it to pull from a (unchanging) tag instead. See here for a list of available tags which correspond to neuron versions; generally you would pick the latest tag and put it in publish.yml. Then, your site will continue use the same neuron version, regardless of upstream development changes.
You will need a GitHub paid plan to publish private repositories. Public repositories on the other hand can be published in the GitHub free plan. Cerveau's Premium Plan, when it is ready, will be able to publish public and private repositories.